Natural Broadening From Heisenberg's uncertainty principle: The electron in an excited state is only there for a short time, so its energy cannot have a precise value. Since energy levels are "fuzzy," atoms can absorb photons with slightly different energy, with the probability of absorption declining as the difference in the photon's energy from the "true" energy of the transition increases. The FWHM of natural broadening for a transition with an average waiting time of  t o is given by A typical value of (  ) 1/2 = 2 x A. Natural broadening is usually very small. The profile of a naturally broadened linen is given by a dispersion profile (also called a damping profile, a Lorentzian profile, a Cauchy curve, and the Witch of Agnesi!) of the form (in terms of frequency) where  is the "damping constant."

The Classical Damping Constant For a classical harmonic oscillator, The shape of the spectral line depends on the size of the classical damping constant For - 0 >>  /4 , the line falls off as ( - 0 ) -2 Accelerating electric charges radiate. and is the classical damping constant ( is in cm) The mean lifetime is also defined as T=1/ , where T=4.5 2

Add Quantum Mechanics Define the oscillator strength, f: related to the atomic transition probability B ul : f-values usually tabulated as gf-values. theoretically calculated laboratory measurements solar

Collisional Broadening Perturbations by discrete encounters Change in energy approximated by a power law of the form  E = constant x r -n Perturbations by static ion fields (linear Stark effect broadening) (n=2) Self-broadening - collisions with neutral atoms of the same kind (resonance broadening, n=3) if perturbed atom or ion has an inner core of electrons (i.e. with a dipole moment) (quadratic Stark effect, n=4) Collisions with atoms of another kind (neutral hydrogen atoms) (van der Waals, n=6) Assume adiabatic encounters (electron doesn’t change level) Non-adiabatic (electron changes level) collisions also possible

Approaches to Collisional Broadening Statistical effects of many particles (pressure broadening) –Usually applies to the wings, less important in the core Some lines can be described fully by one or the other Know your lines! The functional form for collisional damping is the same as for radiation damping, but  rad is replaced with  coll Collisional broadening is also described with a dispersion function Collisional damping is sometimes 10’s of times larger than radiation damping

Damping Coefs for Na D

Doppler Broadening Two components contribute to the intrinsic Doppler broadening of spectral lines: –Thermal broadening –Turbulence – the dreaded microturbulence! Thermal broadening is controlled by the thermal velocity distribution (and the shape of the line profile) where v r is the line of sight velocity component The Doppler width associated with the velocity v 0 (where the variance v 0 2 =2kT/m) is and is the wavelength of line center

More Doppler Broadening Combining these we get the thermal broadening line profile: At line center, = 0, and this reduces to Where the line reaches half its maximum depth, the total width is

Thermal + Turbulence The average speed of an atom in a gas due to thermal motion - Maxwell Boltzmann distribution. The most probably speed is given by Moving atoms are Doppler shifted, and individual atoms will absorb light at slightly different wavelengths because of the Doppler shift. Spectral lines are also Doppler broadened by turbulent motions in the gas. The combination of these two effects produces a Doppler-broadened profile: Typical values for  1/2 are a few tenths of an Angstrom. The line depth for Doppler broadening decreases exponentially from the line center.

Combining the Natural, Collisional and Thermal Broadening Coefficients The combined broadening coefficient is just the convolution of all of the individual broadening coefficients The natural, Stark, and van der Waals broadening coefficients all have the form of a dispersion profile: With damping constants (  rad,  2,  4,  6 ) one simply adds them up to get the total damping constant: The thermal profile is a Gaussian profile:

The Voigt Profile The convolution of a dispersion profile and a Gaussian profile is known as a Voigt profile. Voigt functions are tabulated for use on computations In general, the shapes of spectra lines are defined in terms of Voigt profiles Voigt functions are dominated by Doppler broadening at small , and by radiation or collisional broadening at large  For weak lines, it’s the Doppler core that dominates. In solar-type stars, collisions dominate , so one needs to know the damping constant and the pressure to compute the line absorption coefficient For strong lines, we need to know the damping parameters to interpret the line.

Calculating Voigt Profiles Tabulated as the Hjerting function H(u,a) u=  /  D a=( 2 /4  c)/  D =(  /4  )  D Hjertung functions are expanded as: H(u,a)=H 0 (u) + aH 1 (u) + a 2 H 2 (u) + a3H3(u) +… or, the absorption coefficient is

Line Profiles

Plot a Damped Profile